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Point collocation methods using the fast moving least-square reproducing kernel approximation. (English) Zbl 1054.76066

Summary: A pseudo-spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least-square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. We show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder.

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
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[1] Aluru, International Journal for Numerical Methods in Engineering 47 pp 1083– (2000)
[2] Gingold, Monthly Notices of the Royal Astronomical Society 181 pp 275– (1977) · Zbl 0421.76032 · doi:10.1093/mnras/181.3.375
[3] Nayroles, Computational Mechanics 10 pp 307– (1992)
[4] Lu, Computer Methods in Applied Mechanics and Engineering 113 pp 397– (1994)
[5] Liu, International Journal for Numerical Methods in Fluids 20 pp 1081– (1995)
[6] Liu, International Journal for Numerical Methods in Engineering 38 pp 1655– (1995)
[7] Melenk, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996)
[8] Duarte, Computer Methods in Applied Mechanics and Engineering 139 pp 237– (1996)
[9] Liu, Computer Methods in Applied Mechanics and Engineering 143 pp 422– (1996)
[10] Zhu, Computational Mechanics 22 pp 174– (1998)
[11] Atluri, Computational Mechanics 24 pp 348– (1999)
[12] Liu, International Journal for Numerical Methods in Fluids 24 pp 1391– (1997)
[13] Analysis of a meshfree method for the compressible Euler equations, preprint. · Zbl 1108.76054
[14] Günther, Computer Methods in Applied Mechanics and Engineering 190 pp 279– (2000)
[15] Choe, Discrete and Continuous Dynamical Systems-Series B 1 pp 495– (2001)
[16] Meshfree method for the non-stationary incompressible Navier-Stokes equations, preprint.
[17] Fürst, Zeitschrift f??r Angewandte Mathematik und Mechanik 81 pp 403– (2001)
[18] Xiong, International Journal for Numerical Methods in Engineering 51 pp 1089– (2001)
[19] Belytschko, International Journal for Numerical Methods in Engineering 43 pp 785– (1998)
[20] Li, Computational Mechanics 21 pp 28– (1998)
[21] Li, International Journal for Numerical Methods in Engineering 45 pp 251– (1999)
[22] Li, International Journal for Numerical Methods in Engineering 45 pp 289– (1999)
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